Determine all such integers a and b for which one of the roots of 3x^3+ax^2+bx+12=0 is equal to 1 + 3^(1/2).

if 1+sqrt3 is a root then so is 1-sqrt3:
(x-(1+sqrt3))(x-(1-sqrt3))*P(x) = 0
(x^2 - 2x -2)*P(x) = 0
P(x) must be linear in the form: (mx + n). So now you can mutiply out:
(x^2 -2x - 2)*(mx + n).
now equate the two expressions coeficient to coeficient an I got a = -12, b = 6.